Appendix B — Mathematical Reference

A quick-reference summary of the mathematics used throughout this book.

B.1 Linear Algebra

Key Operations

Eigendecomposition

For symmetric $A$: $$A = Q \Lambda Q^\top$$ where $Q$ is orthogonal (columns are eigenvectors) and $\Lambda = \text{diag}(\lambda_1, \ldots, \lambda_n)$.

Solving Linear Systems

$Ax = b$ solved by LU decomposition: $O(n^3)$ complexity.

B.2 Calculus and Optimisation

Taylor Series (second order)

$$f(x + \delta) \approx f(x) + f'(x),\delta + \tfrac{1}{2} f''(x),\delta^2$$

Integration by Parts

$$\int_a^b u,dv = [uv]_a^b - \int_a^b v,du$$

Gradient Descent

$$\theta_{k+1} = \theta_k - \eta \nabla_\theta \mathcal{L}(\theta_k)$$

Convergence guaranteed for $L$-smooth convex $\mathcal{L}$ with $\eta < 1/L$.

B.3 Probability and Statistics

Normal Distribution

$$\phi(x) = \frac{1}{\sqrt{2\pi}} e^{-x^2/2}, \qquad \Phi(x) = \int_{-\infty}^x \phi(t),dt$$

Useful identities:

• $\Phi(-x) = 1 - \Phi(x)$
• $E[e^{\sigma Z}] = e^{\sigma^2/2}$ for $Z \sim N(0,1)$

Log-Normal Distribution

If $X = e^{\mu + \sigma Z}$, $Z \sim N(0,1)$: $$E[X] = e^{\mu + \sigma^2/2}, \qquad \text{Var}(X) = e^{2\mu+\sigma^2}(e^{\sigma^2}-1)$$

Moment Generating Function

$$M_X(t) = E[e^{tX}]$$

For $X \sim N(\mu, \sigma^2)$: $M_X(t) = e^{\mu t + \sigma^2 t^2/2}$.

B.4 Stochastic Calculus

Brownian Motion Properties

• $W_0 = 0$
• $W_t - W_s \sim N(0, t-s)$ for $t > s$
• Independent increments
• Continuous paths (almost surely)

Itô's Lemma

For $f(t, W_t)$ twice differentiable: $$df = \frac{\partial f}{\partial t},dt + \frac{\partial f}{\partial x},dW_t + \frac{1}{2}\frac{\partial^2 f}{\partial x^2},dt$$

Geometric Brownian Motion

$$dS_t = \mu S_t,dt + \sigma S_t,dW_t$$ $$S_t = S_0 \exp!\left[\left(\mu - \tfrac{\sigma^2}{2}\right)t + \sigma W_t\right]$$

Girsanov Theorem

Under measure $\mathbb{Q}$ defined by: $$\frac{d\mathbb{Q}}{d\mathbb{P}} = \exp!\left(-\int_0^T \theta_t,dW_t - \tfrac{1}{2}\int_0^T \theta_t^2,dt\right)$$ the process $\tilde{W}_t = W_t + \int_0^t \theta_s,ds$ is a $\mathbb{Q}$-Brownian motion.

B.5 Black-Scholes Formula Quick Reference

$$C = S,\Phi(d_1) - K e^{-rT}\Phi(d_2)$$ $$P = K e^{-rT}\Phi(-d_2) - S,\Phi(-d_1)$$

$$d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}, \qquad d_2 = d_1 - \sigma\sqrt{T}$$

Greeks

B.6 Standard Normal Table (selected values)

B.7 Key Financial Formulas

Bond Duration and Convexity

$$D = \frac{1}{P}\sum_{i=1}^n \frac{t_i \cdot C_i}{(1+y)^{t_i}}, \qquad \text{Cx} = \frac{1}{P}\sum_{i=1}^n \frac{t_i(t_i+1)\cdot C_i}{(1+y)^{t_i+2}}$$

Nelson-Siegel Yield Curve

$$y(\tau) = \beta_0 + (\beta_1 + \beta_2)\frac{1 - e^{-\tau/\lambda}}{\tau/\lambda} - \beta_2 e^{-\tau/\lambda}$$

Vasicek Short Rate

$$dr_t = \kappa(\theta - r_t),dt + \sigma,dW_t$$

Zero-coupon bond: $P(t,T) = A(t,T)e^{-B(t,T)r_t}$ where: $$B(t,T) = \frac{1 - e^{-\kappa(T-t)}}{\kappa}$$ $$\ln A(t,T) = \left(\theta - \frac{\sigma^2}{2\kappa^2}\right)(B(t,T) - (T-t)) - \frac{\sigma^2}{4\kappa}B(t,T)^2$$

B.8 Numerical Methods Reference

Root-Finding

Bisection — guaranteed convergence, $O(\log_2((b-a)/\varepsilon))$ iterations: $$x_{n+1} = \frac{a_n + b_n}{2}, \quad \text{update } [a,b] \text{ to keep sign change}$$

Newton-Raphson — quadratic convergence near a simple root: $$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$$

Convergence: $|x_{n+1} - x^| \leq C|x_n - x^|^2$ for some $C > 0$.

Brent's method — combines bisection, secant, and inverse quadratic interpolation. Superlinear convergence, guaranteed to converge. Preferred in practice.

Numerical Integration

Composite Simpson's rule (error $O(h^4)$): $$\int_a^b f(x),dx \approx \frac{h}{3}\left[f(x_0) + 4f(x_1) + 2f(x_2) + \cdots + 4f(x_{n-1}) + f(x_n)\right]$$

Gauss-Legendre quadrature — exact for polynomials of degree $\leq 2n-1$ with $n$ nodes: $$\int_{-1}^1 f(x),dx \approx \sum_{i=1}^n w_i f(x_i)$$

5-point nodes and weights on $[-1, 1]$:

Monte Carlo integration (error $O(1/\sqrt{N})$): $$\int_\Omega f(x),dx \approx \frac{|\Omega|}{N}\sum_{i=1}^N f(x_i), \quad x_i \sim \text{Uniform}(\Omega)$$

Finite Differences

B.9 Stochastic Calculus Extended Reference

Itô Integral Properties

For adapted process $H_t$: $$\int_0^T H_t,dW_t \sim N!\left(0, \int_0^T H_t^2,dt\right) \quad \text{(Itô isometry)}$$

$$E!\left[\int_0^T H_t,dW_t\right] = 0$$

Itô's Lemma (General Form)

For $f(t, X_t)$ where $dX_t = \mu_t,dt + \sigma_t,dW_t$: $$df = \left(\frac{\partial f}{\partial t} + \mu_t\frac{\partial f}{\partial x} + \frac{\sigma_t^2}{2}\frac{\partial^2 f}{\partial x^2}\right)dt + \sigma_t\frac{\partial f}{\partial x},dW_t$$

Multidimensional Itô's Lemma

For $f(t, X_t^1, \ldots, X_t^n)$ with $dX_t^i = \mu_t^i,dt + \sum_j \sigma_t^{ij},dW_t^j$: $$df = \frac{\partial f}{\partial t},dt + \sum_i \frac{\partial f}{\partial x_i},dX_t^i + \frac{1}{2}\sum_{i,j}\frac{\partial^2 f}{\partial x_i \partial x_j},d\langle X^i, X^j\rangle_t$$

where $d\langle X^i, X^j\rangle_t = \sum_k \sigma_t^{ik}\sigma_t^{jk},dt$.

Feynman-Kac Formula

If $u(t,x)$ satisfies the PDE: $$\frac{\partial u}{\partial t} + \mu(t,x)\frac{\partial u}{\partial x} + \frac{\sigma^2(t,x)}{2}\frac{\partial^2 u}{\partial x^2} - r,u = 0$$ with terminal condition $u(T,x) = g(x)$, then: $$u(t,x) = E^{\mathbb{Q}}!\left[e^{-r(T-t)}g(X_T),\Big|,X_t = x\right]$$

This connects PDEs (Crank-Nicolson) to Monte Carlo simulation.

Ornstein-Uhlenbeck Process

$$dX_t = \kappa(\theta - X_t),dt + \sigma,dW_t$$

Exact solution: $X_t = \theta + (X_0 - \theta)e^{-\kappa t} + \sigma\int_0^t e^{-\kappa(t-s)},dW_s$

Stationary distribution: $X_\infty \sim N!\left(\theta, \frac{\sigma^2}{2\kappa}\right)$

B.10 Interest Rate Models Reference

Short Rate Models

CIR Model

$$dr_t = \kappa(\theta - r_t),dt + \sigma\sqrt{r_t},dW_t$$

Feller condition for non-negative rates: $2\kappa\theta > \sigma^2$.

Zero-coupon bond: $P(t,T) = A(t,T)e^{-B(t,T)r_t}$ where: $$B(t,T) = \frac{2(e^{\gamma(T-t)}-1)}{(\gamma+\kappa)(e^{\gamma(T-t)}-1)+2\gamma}, \quad \gamma = \sqrt{\kappa^2 + 2\sigma^2}$$

Forward Rate Agreement (FRA)

A FRA on the rate $L(T_1, T_2)$ has payoff at $T_2$: $$N \cdot \delta \cdot (L(T_1, T_2) - K)$$ where $\delta = T_2 - T_1$ (day count fraction). PV at $t < T_1$: $$\text{PV} = N\delta,P(t,T_2),(F(t;T_1,T_2) - K)$$

LIBOR Market Model (BGM)

Forward LIBOR rates $L_k(t)$ for period $[T_k, T_{k+1}]$ under the terminal measure $\mathbb{Q}^{T_N}$: $$dL_k(t) = L_k(t)\left(-\sum_{j=k+1}^{N-1}\frac{\delta_j L_j(t)\sigma_k(t)\cdot\sigma_j(t)}{1+\delta_j L_j(t)}\right)dt + L_k(t)\sigma_k(t)\cdot d\tilde{W}_t$$

B.11 Credit Mathematics

Hazard Rate and Survival Probability

For constant hazard rate $\lambda$: $$Q(\tau > t) = e^{-\lambda t}, \qquad Q(\tau \leq t) = 1 - e^{-\lambda t}$$

For time-varying $\lambda(t)$: $$Q(\tau > t) = \exp!\left(-\int_0^t \lambda(s),ds\right)$$

CDS Pricing (Simplified)

Par spread $s$ satisfies: $$s = \frac{(1-R)\int_0^T P(0,t),dQ(\tau \leq t)}{\int_0^T P(0,t),Q(\tau > t),dt}$$

where $R$ is the recovery rate and $P(0,t)$ is the risk-free discount factor.

Merton Model

Equity as a call on firm assets: $$E_0 = V_0\Phi(d_1) - De^{-rT}\Phi(d_2)$$ $$d_1 = \frac{\ln(V_0/D) + (r + \sigma_V^2/2)T}{\sigma_V\sqrt{T}}, \quad d_2 = d_1 - \sigma_V\sqrt{T}$$

Risk-neutral default probability: $\Phi(-d_2)$.

B.12 Risk Measures

Value at Risk

For loss $L$ with CDF $F_L$: $$\text{VaR}_\alpha = F_L^{-1}(\alpha) = \inf{l : F_L(l) \geq \alpha}$$

For $L \sim N(\mu, \sigma^2)$: $\text{VaR}_\alpha = \mu + \sigma\Phi^{-1}(\alpha)$.

Expected Shortfall (CVaR)

$$\text{ES}\alpha = E[L \mid L \geq \text{VaR}\alpha] = \frac{1}{1-\alpha}\int_\alpha^1 \text{VaR}_u,du$$

For $L \sim N(\mu, \sigma^2)$: $$\text{ES}_\alpha = \mu + \sigma\frac{\phi(\Phi^{-1}(\alpha))}{1-\alpha}$$

ES is a coherent risk measure (satisfies subadditivity); VaR is not.

Coherent Risk Measure Axioms

A risk measure $\rho$ is coherent if:

1. Monotonicity: $X \leq Y \Rightarrow \rho(X) \geq \rho(Y)$
2. Subadditivity: $\rho(X+Y) \leq \rho(X) + \rho(Y)$
3. Positive homogeneity: $\rho(\lambda X) = \lambda\rho(X)$ for $\lambda > 0$
4. Translation invariance: $\rho(X + c) = \rho(X) - c$

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